# Siegel disks of the tangent family

## TL;DR

This paper investigates the properties of Siegel disks in tangent family functions, establishing conditions for unboundedness and constructing examples with bounded disks using quasiconformal surgery.

## Contribution

It provides a new criterion linking boundary asymptotic values to unbounded Siegel disks and demonstrates construction methods for bounded disks.

## Key findings

- Unbounded Siegel disks contain boundary asymptotic values.
- A forward invariant Siegel disk is unbounded iff it contains an asymptotic value.
- Constructed examples of functions with bounded Siegel disks.

## Abstract

We study Siegel disks in the dynamics of functions from the tangent family. In particular, we prove that a forward invariant Siegel disk is unbounded if and only if it contains at least one asymptotic value on the boundary. Our argument is elementary and function-theoretic. Moreover, by using quasiconformal surgery we also construct functions in the above family with bounded Siegel disks.